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Math  /  Trigonometry

Question\#5 Shari and Leo are standing 36 m apart and facing each other. If Shari looks upward, the angle of elevation of a hot air balloon in the sky is π4\frac{\pi}{4} radians and from Leo the angle of elevation is π3\frac{\pi}{3} radians. Calculate the height of the hot air balloon from the ground.

Studdy Solution

STEP 1

1. Shari and Leo are standing on level ground.
2. The distance between Shari and Leo is 36 36 meters.
3. The angles of elevation from Shari and Leo to the hot air balloon are π4\frac{\pi}{4} and π3\frac{\pi}{3} radians, respectively.

STEP 2

1. Use trigonometry to express the height of the balloon in terms of the distance from Shari.
2. Use trigonometry to express the height of the balloon in terms of the distance from Leo.
3. Set up an equation using the total distance between Shari and Leo.
4. Solve for the height of the balloon.

STEP 3

Use the tangent function for Shari's angle of elevation:
tan⁡(π4)=hx\tan\left(\frac{\pi}{4}\right) = \frac{h}{x}
where h h is the height of the balloon and x x is the horizontal distance from Shari to the point directly below the balloon.

STEP 4

Use the tangent function for Leo's angle of elevation:
tan⁡(π3)=h36−x\tan\left(\frac{\pi}{3}\right) = \frac{h}{36 - x}
where 36−x 36 - x is the horizontal distance from Leo to the point directly below the balloon.

STEP 5

Set up the equations from Steps 1 and 2:
tan⁡(π4)=1⇒h=x\tan\left(\frac{\pi}{4}\right) = 1 \Rightarrow h = x
tan⁡(π3)=3⇒h=3(36−x)\tan\left(\frac{\pi}{3}\right) = \sqrt{3} \Rightarrow h = \sqrt{3}(36 - x)
Equate the two expressions for h h :
x=3(36−x)x = \sqrt{3}(36 - x)

STEP 6

Solve for x x :
x=3(36−x)x = \sqrt{3}(36 - x)
x=363−3xx = 36\sqrt{3} - \sqrt{3}x
x+3x=363x + \sqrt{3}x = 36\sqrt{3}
x(1+3)=363x(1 + \sqrt{3}) = 36\sqrt{3}
x=3631+3x = \frac{36\sqrt{3}}{1 + \sqrt{3}}
Now, substitute back to find h h :
h=x=3631+3h = x = \frac{36\sqrt{3}}{1 + \sqrt{3}}
Rationalize the denominator:
h=363(1−3)(1+3)(1−3)h = \frac{36\sqrt{3}(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}
h=363−1081−3h = \frac{36\sqrt{3} - 108}{1 - 3}
h=363−108−2h = \frac{36\sqrt{3} - 108}{-2}
h=54−183h = 54 - 18\sqrt{3}
The height of the hot air balloon from the ground is:
54−183 meters \boxed{54 - 18\sqrt{3}} \, \text{meters}

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